The source of leaks can be very difficult to track down. Water runs downhill, except when it is between layers of material. Then it can seep uphill, sideways, through other layers, where it then can drip to another surface to repeat the process almost indefinitely. Leak paths are routinely more perplexing than the Warren Commission's Magic Bullet Theory.
A photo from the top of the roof would be very helpful. But in general, walk the roof and look for the leakiest-looking areas and replace the shingles. Then look for gaps, cracks, and holes and fill them with roofing goo (this one can be used while it is raining).
Carefully scrutinize all valleys for cracks and punctures.
A common source of leaks is through-the-roof devices. Check specifically the seals on the side and top vents for water entrapment. For example, this photo highlights problem areas:
Sealing around through-the-roof devices should be done with the roof dry and using silicon caulk. Even the goo on there must slope away from the fixture, otherwise it creates a "leak reservoir".
I once had a leak like this, which didn't look like it could be source of the near-waterfall underneath. But it was.